Question: Solve for $x$ and $y$ using elimination. $\begin{align*}7x+6y &= -6 \\ 7x+8y &= 6\end{align*}$
We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}-7x-6y &= 6\\ 7x+8y &= 6\end{align*}$ Add the top and bottom equations. $2y = 12$ Divide both sides by $2$ and reduce as necessary. $y = 6$ Substitute $6$ for $y$ in the top equation. $7x+6( 6) = -6$ $7x+36 = -6$ $7x = -42$ $x = -6$ The solution is $\enspace x = -6, \enspace y = 6$.